Method of obtaining an antenna gain

ABSTRACT

Method of obtaining a gain function by means of an array of antennae and a weighting of the signals received or to be transmitted by vectors ({overscore (b)}) of complex coefficients, referred to as weighting vectors, according to which, a reference gain function being given, the said reference gain function is projected orthogonally onto the sub-space of the gain function generated by the said weighting vectors of the space of the gain functions, provided in advance with a norm, and a weighting vector generating the reference gain function thus projected is chosen as the optimum weighting vector.

[0001] The present invention concerns in general terms a method ofobtaining an antenna gain function. More particularly, the presentinvention relates to a method of obtaining an antenna gain for a basestation in a mobile telecommunication system. It makes it possible toobtain an antenna gain function, in transmission or reception mode,which is invariant by change of frequency.

[0002] The formation of channels or the elimination of interferingsignals is well known in the field of narrow-band antenna processing.Both of these use an array of antennae, generally linear and uniform(that is to say with a constant pitch) and a signal weighting module.More precisely, if it is wished to form a channel in reception mode, thesignals received by the different antennae are weighted by means of aset of complex coefficients before being added. Conversely, if it iswished to form a channel in transmission mode, the signal to betransmitted is weighted by a set of complex coefficients and the signalsthus weighted are transmitted by the different antennae.

[0003]FIG. 1 illustrates a known device for obtaining antenna gain intransmission and reception mode. The device comprises an array ofantennae (10 ₀),(10 ₁), . . . ,(10 _(N-1)), a transmission weightingmodule (11) and a reception weighting module (15). The signals receivedby the different antennae, (x_(i)), i=0 . . . N-1 are weighted at (13₀),(13 ₁) . . . ,(13 _(N-1)) by a set of complex coefficients (b_(ui)),i=0, . . . ,N-1 before being added at (14) in order to give a signalR_(u). Conversely, a signal to be transmitted S_(d) is weighted at(12₀),(12₁) . . . ,(12_(N-1)) by means of a set of complex coefficients(b_(di)), i=0, . . ,N-1, before being transmitted by the differentantennae.

[0004] If respectively the vector of the signals received and the vectorof the weighting coefficients is denoted {overscore (x)}=(x0,x1, . . .,xN-1)^(τ) and {overscore (b^(u))}=(bu0,bu1, . . . ,buN-1)^(τ), it ispossible to write:

Ru={overscore (b)}_(u){overscore (x)}  (1)

[0005] The complex gain (or the complex gain function of the antenna) inreception mode can be written: $\begin{matrix}{{G\left( {\overset{\_}{b_{u}},\theta} \right)} = {{{\overset{\_}{b_{u}}}^{T} \cdot \overset{\_}{e_{u}\theta}} = {\sum\limits_{i = 0}^{N - 1}\quad {b_{ui} \cdot {\exp \left( {- {j\phi}_{i}} \right)}}}}} & (2)\end{matrix}$

[0006] where euθ represents the vector {overscore (x)} corresponding toa flat wave arriving at an angle of incidence θ, and

φ=(2 πd/λ).i.sin(θ)=(2 πdf/c).i.sin(θ)  (3)

[0007] is the difference in operation between consecutive antennae for auniform linear array of pitch d, λ and f being respectively thewavelength and the frequency of the flat wave in question;

φi=2πRΔθ/λsin(θ-θ_(i))=2πRfΔθ/c.sin(θ-θ_(i))  (4)

[0008] for a circular array where θ_(i) is the angle between a referenceaxis and the normal to the antenna of index i, R the radius of curvatureof the array, Δθ is the angular difference between two consecutiveantennae in the array.

[0009] Likewise the complex gain (or the complex gain function) intransmission mode can be written: $\begin{matrix}{{G\left( {\overset{\_}{b_{d}},\theta} \right)} = {{{\overset{\_}{b_{d}}}^{T} \cdot \overset{\_}{e_{d}\theta}} = {\sum\limits_{i = 0}^{N - 1}\quad {b_{di} \cdot {\exp \left( {j\phi}_{i} \right)}}}}} & (5)\end{matrix}$

[0010] with the same conventions as those adopted above and where{overscore (edθ)} designates the vector {overscore (x)} corresponding toa flat wave transmitted in the direction θ. The weighting vectors inreception and transmission mode respectively will be called {overscore(bu)} and {overscore (bd)}.

[0011] Clearly, the antenna gain in transmission or reception modedepends on the frequency of the signal in question. There are howevermany situations in which the antenna gain must remain unchanged whateverthe frequency of the signal. For example, in so-called FDD (FrequencyDivision Duplex) mobile telecommunication systems, where the frequencyused on the downlink, that is to say from the base station to the mobilestation, differs from that used on the uplink. Similarly, infrequency-hopping radar systems, it is necessary to ensure theinvariance of the gain function, notably in order to aim a transmissionor reception beam in a given direction or to eliminate the interferencecoming from a given direction, whatever the frequency used.

[0012] In more general terms, it is desirable to be able to obtain, fora given signal frequency, an antenna gain function which is as close aspossible, in the sense of a certain metric, to a reference gainfunction. The reference gain function can notably be a gain functionobtained at a given frequency which it is sought to approximate to thegreatest possible extent during transmission or reception at anotherfrequency.

[0013] The aim of the invention is to propose a method of obtaining again function making it possible, for a given signal frequency, toapproach a reference gain function as closely as possible.

[0014] A subsidiary aim of the invention is to propose a method for bestapproaching an antenna gain function obtained at a given frequency whenthe network is transmitting or receiving at another frequency.

[0015] To this end, the invention is defined by a method of obtaining again function by means of an array of antennae and a weighting of thesignals received or to be transmitted by vectors ({overscore (b)}) of Ncomplex coefficients, referred to as weighting vectors, N being thenumber of antennae in the array, according to which, a reference gainfunction being given, the said reference gain function is projectedorthogonally onto the sub-space of the gain functions generated by thesaid weighting vectors of the space of the gain functions, provided inadvance with a norm, and a weighting vector generating the referencegain function thus projected is chosen as the optimum weighting vector.

[0016] The gain functions are preferably represented by vectors({overscore (G)}), referred to as gain vectors, of M complex samplestaken at M distinct angles, defining sampling directions and belongingto the angular range covered by the array, the space of the gainfunctions then being the vector space C^(M) provided with the Euclidiannorm and that, for a given frequency (f), the reference gain vector isprojected on the vector sub-space (Imf) of the gain vectors generated bythe array operating at the said frequency in order to obtain the saidoptimum weighting vector.

[0017] Advantageously, M is chosen such that M>πN.

[0018] According to one example embodiment, the sampling angles aredistributed uniformly in an angular range covered by the array.

[0019] The reference gain vector can be obtained by sampling thereference gain function after anti-aliasing filtering.

[0020] The gain vectors ({overscore (G)}) being the transforms by alinear application (h_(s) ^(f)) of C^(N) in C^(M) weighting vectors ofthe array and H_(f) being the matrix, of size M×N, of the said linearapplication of a starting base of C^(N) in an arrival base C^(M,), thesaid optimum weighting vector, for a given frequency f, is preferablyobtained from the reference gain vector {overscore (G)} as {overscore(b)}=H⁺ _(f).{overscore (G)} where H⁺ _(f)=(H*^(T) _(f).H_(f))⁻¹.H*^(T)_(f) is the pseudo-inverse matrix of the matrix H_(f) and where H*_(f)^(T) the conjugate transpose of the matrix H_(f).

[0021] The said starting base being that of the vectors {overscore(e)}_(k), k=0, . . ,N-1, such that {overscore (e)}_(k)=(ek,0,ek,1, . .,ek,N-1)^(T) with$e_{k,i} = {\exp \left( {{j \cdot \frac{2\pi \quad {fd}}{c} \cdot i \cdot \sin}\quad \theta_{k}} \right)}$

[0022] and θ_(k)=kπ/N k=−(N-1)/2, . . . ,0, . .,(N-1)/2 and the arrivalbase being the canonical base, the matrix H_(f) then has the components:$H_{pq} = {{\exp \left( {{j\left( {N - 1} \right)}{\Psi_{pq}/2}} \right)} \cdot \frac{\sin \left( {N\quad {\Psi_{pq}/2}} \right)}{\sin \left( {\Psi_{pq}/2} \right)}}$

[0023] with ψ_(pq)=πη(sin(ρπ/N)-sin(qπ/M)) and η=f/f₀ with f₀=c/2d, dbeing the pitch of the array.

[0024] If the reference gain vector is obtained by sampling the gainfunction generated at a first operating frequency f₁ of the array by afirst weighting vector {overscore (b₁)}, the optimum weighting gainvector for a second frequency f₂ is obtained by {overscore (b₂)}=H⁺_(f2).H_(f1){overscore (b₁)}.

[0025] The frequency f₁ of operation of the array is for example thefrequency of an uplink between a mobile terminal and a base station in amobile telecommunication system and the frequency f₂ of operation of thearray is for example the frequency of a downlink between the said basestation and the said mobile terminal.

[0026] The characteristics of the invention mentioned above, as well asothers, will emerge more clearly from a reading of the followingdescription given in relation to the accompanying figures, amongstwhich:

[0027]FIG. 1 depicts schematically a known device for obtaining anantenna gain function;

[0028]FIG. 2 depicts schematically a device for obtaining an antennagain function according to one example embodiment of the invention.

[0029] A first general idea at the basis of the invention is to bestapproximate a reference gain function by virtue of a linear combinationof base functions.

[0030] A second general idea at the basis of the invention is to samplethe reference gain function and to best approximate the series ofsamples obtained by means of a linear combination of base vectors.

[0031] The first embodiment of the invention consists of approximatingthe reference gain function by means of a linear combination of basefunctions.

[0032] Let h be the linear application of C^(N) in the vector space F ofthe complex functions defined on [−π/2,π/2] (or [−π,π]) which associateswith any vector {overscore (b)} of complex numbers the functionh({overscore (b)}) such that h({overscore (b)})(θ)=G({overscore (b)},θ)where G is a complex gain function in transmission or reception mode asdefined at (2) or (5). C^(N) being a vector space of dimension N on C,the image of C^(N) by h is a vector sub-space of F of dimension at mostequal to N, which will be denoted Im_(f) to emphasise that the imagedepends on the frequency f in question in expression (2) or (5).

[0033] Let G be a reference complex gain function, the problem is tofind the weighting vector {overscore (b)} such that h({overscore (b)})is as close as possible to G in the sense of a certain metric. For auniform linear array, the metric corresponding to the scalar product onF w₁ ⋅ w₂ = ∫_(−π/2)^(π/2)w₁(θ) ⋅ w₂^(′)(θ)cos (θ)  θ

[0034] and therefore to the norm ||w||²=∫_(π/2) ^(π/2)|w(θ)|² cosθ.dθ ischosen. The case of the circular array can be dealt within a similarmanner (the chosen norm does not then include the term cos(θ)). Thespace F₂ of the functions of F of bounded norm is itself a vector spacenormed by the above norm. If G is an element of F₂, the element of thesub-space Im_(f) closest to G is then the projection of G onto thissub-space.

[0035] If the vector sub-space corresponding to the inherent frequencyof the array is considered to be Im_(f0), it is possible to demonstratethat the functions e_(k)(θ), k=0, . . . ,N-1 defined by:

[0036] ek(θ)=h({overscore (bk)})(θ)=G({overscore (bk)},θ), where{overscore (bk)} is the vector of components bki=exp(j.2πki/N), areorthogonal. Being N in number, they therefore form a base of Im_(f0). Inmore general terms, it can be shown that, if two vectors {overscore (b)}and {overscore (b′)} are orthogonal, that is to say are such that{overscore (bb′)}={overscore (b)}^(T){overscore (b′)}=0, the functionsh({overscore (b)}) and h({overscore (b′)}) of Im_(f0) are orthogonal.This is because: $\begin{matrix}{{{h\left( \overset{\_}{b} \right)} \cdot {h\left( \overset{\_}{b^{\prime}} \right)}} = {{\sum\limits_{i = 0}^{N - 1}\quad {\sum\limits_{i^{\prime} = 0}^{N - 1}\quad {{b_{i} \cdot b_{i^{\prime}}^{\prime}}{\int_{{- \pi}/2}^{\pi/2}{{\exp \left( {{j\left( {i - i^{\prime}} \right)}{\phi (\theta)}} \right)}{\cos (\theta)}{\theta}}}}}} = {\sum\limits_{i = 0}^{N - 1}\quad {\sum\limits_{i^{\prime} = 0}^{N - 1}\quad {{b_{i} \cdot b_{i^{\prime}}^{\prime} \cdot \sin}\quad {c\left( {{\pi\eta}\left( {i - i^{\prime}} \right)} \right)}}}}}} & (6)\end{matrix}$

[0037] with φ(θ)=2πfd/c.sinθ=πηsinθ where η=f/f₀≦1, is the ratio of thefrequency used at the maximum frequency f₀=c/2d, which can resolve thearray without ambiguity, which will be referred to as the naturalfrequency of the array, and where sinc. is the cardinal sine function.For η=1, the terms below the sum signs of the second member of equation(6) are zero if i≠i′ and therefore the second member is equal to zero ifthe vectors {overscore (b)} and {overscore (b′)} are orthogonal.

[0038] Consider now the general case of a frequency f≦f₀. Let e_(k)(θ),k=0, . . . ,N-1 be an orthogonal base of Im_(f). By definition,e_(k)(θ)=h({overscore (bk)})(θ)=G({overscore (b_(k))},θ) where{overscore (b_(k) )} is a vector of C^(N). Consider now a gain functionG of F₂. It can be projected onto the vectors e_(k)(θ). If λ_(k)=G.e_(k)is written, then the vector of$C^{N},{\overset{\_}{b_{G}} = {\sum\limits_{k = 0}^{P - 1}\quad {\lambda_{k}\overset{\_}{b_{k}}}}}$

[0039] is such that h({overscore (b_(G))}) best approximates thefunction G.

[0040] The second embodiment of the invention consists of approximatinga vector of samples of the reference gain function by means of a linearcombination of base vectors.

[0041] Let G₀(θ) be the antenna gain function obtained without weightingfor a linear uniform array, it is easily shown that: $\begin{matrix}{{{G_{0}(\theta)}} = {{\frac{\sin \left( {N\quad {\phi/2}} \right)}{\sin \left( {\phi/2} \right)}\quad {with}\quad \phi} = {2\pi \quad {{{fd}/c} \cdot \sin}\quad \theta}}} & (7)\end{matrix}$

[0042] This function has zeros for the values φk=2kπ/N, k integernon-zero such that φkε[−π,π[ that is to say in the directions for whichsinθ_(k)=k.c/Nfd, when this expression has a direction. The phasedifference between two consecutive zeros of the gain diagram is constantand is equal to Δφ=2π/N. The angular difference between two consecutivezeros of the diagram varies in terms of Arcsin., a function whosederivative is increasing on [−1,1] and is therefore at a minimum for theangular difference between the first and second zeros. It is thereforebounded by Δθ_(min)=c/Nfd if N is sufficiently great. It will be assumedthat the frequencies used are less than f₀ where f₀ is the naturalfrequency of the array. It can be concluded from this that the spectrumof the function G₀(θ) has a support bounded by 1/Δ θ_(min)=N/2.

[0043] In more general terms, let G(θ) be the antenna gain functionobtained by means of a weighting vector {overscore (b)}. G can beexpressed as the Fourier transform (FT) (in reception mode) or theinverse Fourier transform (in transmission mode) of the complexweighting distribution of the antenna, namely:${b(x)} = {\sum\limits_{i = 0}^{N - 1}\quad {b_{i} \cdot {\delta \left( {x - x_{i}} \right)}}}$

[0044] with xi=i.d; this gives: G_(b1)(θ)=B(sinθ) withB(u) = ∫_(−∞)^(+∞)b(x)exp (−j2πux/λ)  x

[0045] and likewise G_(d)(θ)=B′(sinθ) withB^(′)(u) = ∫_(−∞)^(+∞)b(x)exp (j2πux/λ)  x.

[0046] The function b(x) being delimited by N.d, the difference betweentwo zeros of the function B or B′ is at least λ/N.d and therefore evenmore so 2/N. Given the increase in the derivative of the functionArcsin. the minimum difference between two zeros of the function G is2/N. The function G therefore has a spectrum delimited by N/2.

[0047] According to the Shannon sampling theorem, it is concluded thatit is possible to reconstitute the function G(θ) if sampling is carriedout at a frequency greater than the Nyquist frequency, i.e. N. In otherwords, for an angular range [−π/2,π/2], at a minimum M>π.N samples arenecessary, where M is integer. In practice K.N samples can be taken withK integer, K≧4.

[0048] For a circular array, it can be shown that 1/Δθ_(min)=N and theangular range being [−π,π], M (M>π.N and M integer) angularlyequidistributed samples also suffice to reconstitute the function G(θ).

[0049] In the general case of the sampling of any gain function G(θ), itis necessary to previously filter G(θ) by means of an anti-aliasingfilter before sampling it. It then suffices to take M samples of thefiltered diagram over the whole of the angular range in order toreconstitute the filtered diagram.

[0050] Let (g_(k)), k=0, . . ,M-1 be the samples of the complex diagram,possibly filtered by an anti-aliasing filtering if necessary, that is tosay g_(k)=G′(θ_(k)) where the θ_(k) are M angles equidistributed over[−π/2,π/2] or [−π,π] and where it is assumed that G′ was the filteredversion of the reference complex diagram.

[0051] It is now possible to define a linear application, h^(f) _(s) ofC^(N) in C^(M) which makes the vector h_(s) ^(f)({overscore(b)})={overscore (G)}=(g₀,g₁, . . ,g_(M-1))^(T), whereg_(k)=G({overscore (b)},θk), correspond to any vector {overscore (b)}.The image of C^(N) by h^(f) _(s), is a vector sub-space of C^(M) ofdimension at most equal to N, which will be noted Im_(f). If a base ofC^(N) is chosen, for example the canonical base, and a base of C^(M),the linear application h^(f) _(s) can be expressed by a matrix H_(f) ofsize M×N which is at most of rank N.

[0052] Let {overscore (G)} be any gain vector corresponding to a sampledgain function. The problem is to find a vector {overscore (b)} such thath_(s) ^(f)({overscore (b)}) is the closest to {overscore (G)} in thesense of a certain metric. The Euclidian norm on C^(M), namely${{\overset{\_}{G}}^{2} = {\sum\limits_{k = 0}^{M - 1}\quad {g_{k}}^{2}}},$

[0053] will be taken as the norm. If it exists, the sought-for vector{overscore (b)} is then such that h_(s) ^(f)(b)={overscore (G)}_(p)where {overscore (G)}_(p) is the orthogonal projection of the vector{overscore (G)} onto Im_(f). If the matrix H_(f) is of rank N, thesought-for vector {overscore (b)} exists and can be written:

{overscore (b)}=H_(f) ⁺.{overscore (g)}  (8)

[0054] where H_(f) ⁺=(H*_(f) ^(T).H_(f))⁻¹.H*_(f) ^(T) is thepseudo-inverse matrix of the matrix H_(f) with transposed H*_(f) ^(T)the conjugate of the matrix H_(f).

[0055] In the discrete case as in the continuous case, the referencegain function (sampled in the discrete case) is projected onto thesub-space generated by the functions (continuous case) or the vectors(discrete case) associated with the array weighting vectors.

[0056] In order to express the matrix H_(f), it is necessary to agree abase of the starting space and a base of the arrival space. It ispossible to choose as a base of C^(M) the canonical base and as a baseof C^(N) a base adapted to the description of the flat waves offrequency f. Consider the distinct vectors {overscore (e)}_(k), k=0, . .,N-1, such that {overscore (e)}_(k)=(ek,0ek,1, . . ,ek,N-1)^(T) with$e_{k,i} = {{\exp \left( {{j \cdot \frac{2\pi \quad {fd}}{c} \cdot i \cdot \sin}\quad \theta_{k}} \right)} = {\exp \left( {j\quad {\pi \cdot \eta \cdot i \cdot \sin}\quad \theta_{k}} \right)}}$

[0057] with η=f/f₀ and where the θ_(k) belong to the interval[−π/2,π/2]. The vectors ek are the weighting vectors of the array makingit possible to form beams in the directions θ_(k). The vectors{overscore (e)}_(k) form a base if the determinant of the coordinates ofthe {overscore (e)}_(k) in the canonical base of C^(N) is non-zero. Thisdeterminant is a Vandermonde determinant which is equal to Π_(p≠q)(exp(jφp)−exp(jφq)) with φκ=πηsin θκ. This determinant is cancelled outif and only if there are two angles θ_(p) and θ_(q) such thatsinθ_(p)-sinθ_(q)=2/η. In other words, for η<1 the N vectors {overscore(e)}_(k) always form a base, and for η=1 only the case θ_(p)=−θ_(q)=π/2is excluded. The directions can, for example, be chosen so as to beequidistributed, that is to say such that θ_(k)=kπ/N with k=−(N-1)/2, .. . ,0, . . ,(N-1)/2. In this case, the matrix H_(f) has as itscomponents: $\begin{matrix}{{{H_{pq} = {\sum\limits_{i = 0}^{N - 1}\quad {{\exp \left( {j\quad {{\pi\eta} \cdot i \cdot {\sin \left( {p\quad {\pi/N}} \right)}}} \right)}{\exp \left( {{- j}\quad {{\pi\eta} \cdot i \cdot {\sin \left( {q\quad {\pi/M}} \right)}}} \right)}}}}\quad {{or}\text{:}}\quad {H_{pq} = {{\sum\limits_{i = 0}^{N - 1}\quad {\exp \left( {j\quad {{\pi\eta} \cdot i \cdot \left\lbrack {{\sin \left( {p\quad {\pi/N}} \right)} - {\sin \left( {q\quad {\pi/M}} \right)}} \right\rbrack}} \right)}} = {{\exp \left( {{j\left( {N - 1} \right)}{\Psi_{pq}/2}} \right)} \cdot \frac{\sin \left( {N\quad {\Psi_{pq}/2}} \right)}{\sin\left( \quad {\Psi_{pq}/2} \right)}}}}{{{with}\quad \Psi_{pq}} = {{\pi\eta}\left( {{\sin \left( {{p\pi}/N} \right)} - {\sin \left( {q\quad {\pi/M}} \right)}} \right)}}}\quad} & (9)\end{matrix}$

[0058] Alternatively, it is possible to choose as a starting baseanother base adapted to the frequency f, the one formed by the vectors{overscore (e′)}k, such that e′k,i=exp(j π.η.i.sinθ_(k)) withsinθ_(k)=2k/ηN and k=−(N-1)/2, . . . ,0, . . ,(N-1)/2. These vectorsexist if |sinθ_(k)|≦1, ∀k, that is to say for η>1-1/N and in this casethe vectors {overscore (e′)}k form a base which has the advantage ofbeing orthogonal.

[0059] Alternatively, it is possible to choose as a starting base thecanonical base of C^(N), which has the advantage of not depending on thefrequency. In this case, the matrix H′_(f) expressed in this base iswritten:

H′_(f)=H_(f).T⁻¹   (10)

[0060] where T is the matrix of the coordinates of {overscore (e)}_(k)in the canonical base, that is to say T_(pp′)=exp(jπpsin(p′/N)). It wasseen above that this matrix had a non-zero Vandermonde determinant andwas consequently reversible.

[0061] Whatever the chosen base, consider now a gain function obtainedat a first frequency f₁,f₁≦f₀ and {overscore (G₁)}=h_(s)^(f1)({overscore (b₁)}) the vector of the samples associated with thisgain function. Let there be a second working frequency f₂, f₂≦f₀.{overscore (G₁)} belonging to C^(M), if the matrix H_(f2) is of rank N,it is possible to find a vector {overscore (b₂)} such that h_(s)^(f2)({overscore (b₂)}) is the projection of h_(s) ^(f1)({overscore(b₁)}) onto Imf₂. The vector {overscore (b₂)} is obtained by means ofthe matrix equation:

{overscore (b₂)}=H _(f2) ⁺ .H _(f1 {overscore (b 1)})  (11)

[0062] This equation makes it possible in particular to obtain, at asecond working frequency, a sampled gain diagram which is as close topossible to the one, referred to as the reference one, obtained at afirst working frequency.

[0063] Equation (11) advantageously applies to the array of a basestation in a mobile telecommunication system operating in FDD. Equation(10) makes it possible to directly obtain the weighting vector to beapplied for the “downlink” transmission at a frequency f_(d) on theweighting vector relating to the “uplink” transmission at a frequencyf_(u). For the paired frequencies f_(d) and f_(u), it is then possibleto write:

{overscore (bd)}=H _(d) ⁺ .Hu{overscore (bu)}  (12)

[0064] The base station can thus direct transmission beams to the mobileterminals using a gain function optimised for the reception of thesignals transmitted by these terminals.

[0065]FIG. 2 depicts an example of an embodiment implementing the secondembodiment. The device comprises a transmission weighting module (31)and a reception weighting module (35) with a structure identical to thatof the modules (11) and (15) respectively. The module (35) is associatedwith a module (36) supplying the complex coefficients for the formationof reception channels and/or the elimination of signals in theinterference directions. The module (36) determines, in a manner knownper se, a weighting vector {overscore (bu)} which maximises the signalreceived in the useful direction or directions and minimises it in theinterference directions. Advantageously bu is calculated adaptively fromthe signals received by the different antennae. The vector is on the onehand used by the reception weighting module (35) and on the other handtransmitted to a projection and inversion module (32) determining thevector {overscore (bd)} from equation (12). The vector {overscore (bd)}is used for weighting the signals to be transmitted in the module (31).As seen above, the transmission gain diagram at frequency f_(u) willminimise the difference, in the sense of the Euclidian distance, betweenthe transmission gain vector {overscore (Gd)} and the reception gainvector {overscore (Gu)}.

[0066] Although the invention has been essentially described, forreasons of simplicity of presentation, in the context of a uniformlinear array, it can apply to any type of antenna array and notably to acircular array.

1. Method of obtaining a gain function by means of an array of antennaeand a weighting of the signals received or to be transmitted by vectors({overscore (b)}) of N complex coefficients, referred to as weightingvectors, N being the number of antennae in the array, characterised inthat, a reference gain function being given, the said reference gainfunction is projected orthogonally onto the sub-space of the gainfunctions generated by the said weighting vectors of the space of thegain functions, previously provided with a norm, and in that there ischosen, as the optimum weighting vector, a weighting vector generatingthe reference gain function thus projected.
 2. Method of obtaining areference gain function according to claim 1, characterised in that thegain functions are represented by vectors ({overscore (G)}), referred toas gain vectors, of M complex samples taken at M distinct angles,defining sampling directions and belonging to the angular range coveredby the array, the space of the gain functions then being the vectorspace C^(M) provided with the Euclidian norm, and in that, for a givenfrequency (f), the reference gain vector is projected onto the vectorsub-space (Imf) of the gain vectors generated by the array operating atthe said frequency in order to obtain the said optimum weighting vector.3. Method of obtaining a reference gain function according to claim 2,characterised in that M is chosen such that M>πN.
 4. Method of obtaininga reference gain function according to claim 2 or 3, characterised inthat the sampling angles are uniformly distributed in the angular rangecovered by the array.
 5. Method of obtaining a reference gain functionaccording to claim 2, characterised in that the reference gain functionis obtained by sampling the reference gain function after anti-aliasingfiltering.
 6. Method of obtaining a reference gain function according toone of claims 2 to 5, characterised in that, the gain vectors({overscore (G)}) being the transforms by a linear application (h_(s)^(f)) of C^(N) in C^(M) of the weighting vectors of the array and H_(f)being the matrix, of size M×N, of the said linear application of astarting base of C^(N) in an arrival base C^(M), the said optimumweighting vector, for a given frequency f, is obtained from thereference gain vector {overscore (G)} as {overscore(b)}=H+_(f).{overscore (G)} where H+_(f)=(H*_(f) ^(T).H_(f))⁻¹.H*_(F)^(T) is the pseudo-inverse matrix of the matrix H_(f) and where H*_(f)^(T) is the conjugate transpose of the matrix H_(f).
 7. Method ofobtaining a reference gain function according to claim 6, characterisedin that, the said starting base being that of the vectors {overscore(e)}_(k), k=0, .. ,N-1, such that {overscore (e)}_(k)=(ek,0,ek,1, . .,ek,N-1)^(T) with$e_{k,i} = {\exp \left( {{j \cdot \frac{2\pi \quad {fd}}{c} \cdot i \cdot \sin}\quad \theta_{k}} \right)}$

and θ_(k)=kπ/N k=−(N-1)/2, . . . ,0, . . ,(N-1)/2 and the arrival basebeing the canonical base, the matrix H_(f) has as its components:$H_{pq} = {{\exp \left( {{j\left( {N - 1} \right)}{\Psi_{pq}/2}} \right)} \cdot \frac{\sin \left( {N\quad {\Psi_{pq}/2}} \right)}{\sin\left( \quad {\Psi_{pq}/2} \right)}}$

with ψ_(pq)=πη(sin(ρπ/N)−sin(qπ/M)) and η=f/f₀ with f₀=c/2d, d being thepitch of the array.
 8. Method of obtaining a reference gain functionaccording to claim 6 or 7, characterised in that the reference gainvector is obtained by sampling the gain function generated at a firstoperating frequency f₁ of the array by means of a first weighting vector{overscore (b₁)} and in that the optimum weighting gain vector for asecond frequency f₂ is obtained by {overscore (b2)}=H⁺_(f2).H_(f1){overscore (b₁)}.
 9. Method of obtaining a reference gainfunction according to claim 8, characterised in that the operatingfrequency f₁ of the array is the frequency of an uplink between a mobileterminal and a base station in a mobile telecommunication system and inthat the operating frequency f₂ of the array is the frequency of adownlink between the said base station and the said mobile terminal.